Mathematics (Scholarly Articles)

Permanent URI for this collection


Recent Submissions

  • Publication
    Singular travelling waves in soft viscoelastic solids of rate type
    (Elsevier, 2023-09-25) Berjamin, Harold; Destrade, Michel; Saccomandi, Giuseppe; Horizon 2020
    We consider shear wave propagation in soft viscoelastic solids of rate type. Based on objective stress rates, the constitutive model accounts for finite strain, incompressibility, as well as stress- and strain-rate viscoelasticity. The theory generalises the standard linear solid model to three-dimensional volume-preserving motions of large amplitude in a physically-consistent way. The nonlinear equations governing shear motion take the form of a one-dimensional hyperbolic system with relaxation. For specific objective rates of Cauchy stress (lower- and upper-convected derivatives), we study the propagation of acceleration waves and shock waves. Then we show that both smooth and discontinuous travelling wave solutions can be obtained analytically. We observe that the amplitude and velocity of steady shocks are very sensitive to variations of the stress relaxation time. Furthermore, the existence of steady shocks is conditional. Extension of these results to the case of multiple relaxation mechanisms and of the Jaumann stress rate is attempted. The analysis of simple shearing motions is more involved in these cases.
  • Publication
    An atlas of the heterogeneous viscoelastic brain with local power-law attenuation synthesised using Prony-series
    (Elsevier, 2023-09-13) Morrison, Oisín; Destrade, Michel; Tripathi, Bharat B.
    This review addresses the acute need to acknowledge the mechanical heterogeneity of brain matter and to accurately calibrate its local viscoelastic material properties accordingly. Specifically, it is important to compile the existing and disparate literature on attenuation power-laws and dispersion to make progress in wave physics of brain matter, a field of research that has the potential to explain the mechanisms at play in diffuse axonal injury and mild traumatic brain injury in general. Currently, viscous effects in the brain are modelled using Prony-series, i.e., a sum of decaying exponentials at different relaxation times. Here we collect and synthesise the Prony-series coefficients appearing in the literature for twelve regions: brainstem, basal ganglia, cerebellum, corona radiata, corpus callosum, cortex, dentate gyrus, hippocampus, thalamus, grey matter, white matter, homogeneous brain, and for eight different mammals: pig, rat, human, mouse, cow, sheep, monkey and dog. Using this data, we compute the fractional-exponent attenuation power-laws for different tissues of the brain, the corresponding dispersion laws resulting from causality, and the averaged Prony-series coefficients.
  • Publication
    Programmable wrinkling for functionally-graded auxetic circular membranes
    (Elsevier, 2023-07-13) Venkata, Sairam Pamulaparthi; Balbi, Valentina; Destrade, Michel; Accoto, Dino; Zurlo, Giuseppe; Horizon 2020
    Materials with negative Poisson¿s ratio, also known as auxetic materials, display exotic properties such as expansion in all directions under uni-axial tension. For their unique properties, these materials find a broad range of applications in robotic, structural, aerospace, and biomedical engineering. In this work we study the wrinkling behavior of thin and soft auxetic membranes, subjected to edge tractions. We show that spatial inhomogeneities of the Young modulus and of the Poisson ratio can be suitably tailored to produce non-trivial wrinkling patterns, with wrinkled regions that can appear, broaden, merge, and eventually disappear again, as the magnitude of applied tractions is increased monotonically. To model wrinkling in a functionally graded membrane, we employ the mathematically elegant and physically transparent tension field theory, an approximated method that we implement in commercially available software. Beyond unveiling the challenging technological potential to achieve non-standard wrinkling on demand in auxetic membranes, our study also confirms the potential of using tension field theory to study, analytically and numerically, instabilities in functionally graded materials.
  • Publication
    Assessing ISO 18404 standard applicability in the service sector: a qualitative study
    (Emerald, 2024-01-18) Cudney, Elizabeth; Sony, Michael; Ramadan, Mariam Ali; Antony, Jiju; Al Dhaheri, Maha Khalifa; McDermott, Olivia; Cudney, Elizabeth A.
    This research aims to establish the applicability of the International Organisation for Standardisation (ISO) 18404 standard to the service sector, identify any required amendments and identify the critical success factors and barriers to deploying the standard within the service sector. Design/methodology/approach The study used a qualitative approach by interviewing operational excellence (OPEX) professionals who work in the service sector. Findings The findings indicate a significant lack of knowledge about the existence of the standard and a general scepticism regarding the applicability of the current ISO 18404 standard to the service sector. Research limitations/implications Limited examples of the application of ISO 18404 in organisations exist, as only a few organisations have adopted the standard. Therefore, the research focussed on the challenges and obstacles that experienced OPEX professionals perceived could be an issue. Originality/value The study will aid service sector organisations in understanding the standard and, subsequently, determine whether to pursue it as part of an OPEX programme. This research is the first study on the application of ISO 18404 to the service sector.
  • Publication
    Global perspectives on operational excellence: unveiling critical failure factors and sustainable pathways
    (Emerald, 2024-03-14) Antony, Jiju; Sony, Michael; Jayaraman, Raja; Swarnakar, Vikas; Tortorella, Guilherme da Luz; Garza-Reyes, Jose Arturo; Rathi, Rajeev; Gutierrez, Leopoldo; McDermott, Olivia; Lameijer, Bart Alex
    Purpose The purpose of this global study is to investigate the critical failure factors (CFFs) in the deployment of operational excellence (OPEX) programs as well as the key performance indicators (KPIs) that can be used to measure OPEX failures. The study also empirically analyzes various OPEX methodologies adopted by various organizations at a global level. Design/methodology/approach This global study utilized an online survey to collect data. The questionnaire was sent to 800 senior managers, resulting in 249 useful responses. Findings The study results suggest that Six Sigma is the most widely utilized across the OPEX methodologies, followed by Lean Six Sigma and Lean. Agile manufacturing is the least utilized OPEX methodology. The top four CFFs were poor project selection and prioritization, poor leadership, a lack of proper communication and resistance to change issues. Research limitations/implications This study extends the current body of knowledge on OPEX by first delineating the CFFs for OPEX and identifying the differing effects of these CFFs across various organizational settings. Senior managers and OPEX professionals can use the findings to take remedial actions and improve the sustainability of OPEX initiatives in their respective organizations. Originality/value This study uniquely identifies critical factors leading to OPEX initiative failures, providing practical insights for industry professionals and academia and fostering a deeper understanding of potential pitfalls. The research highlights a distinctive focus on social and environmental performance metrics, urging a paradigm shift for sustained OPEX success and differentiating itself in addressing broader sustainability concerns. By recognizing the interconnectedness of 12 CFFs, the study offers a pioneering foundation for future research and the development of a comprehensive management theory on OPEX failures.
  • Publication
    Canceling the elastic Poynting effect with geometry
    (American Physical Society, 2023-05-24) Destrade, Michel; Du, Y.; Blackwell, J.; Colgan, N.; Balbi, V.; Irish Research Council
    The Poynting effect is a paragon of nonlinear soft matter mechanics. It is the tendency (found in all incompressible, isotropic, hyperelastic solids) exhibited by a soft block to expand vertically when sheared horizontally. It can be observed whenever the length of the cuboid is at least four times its thickness. Here we show that the Poynting effect can be easily reversed and the cuboid can shrink vertically, simply by reducing this aspect ratio. In principle, this discovery means that for a given solid, say one used as a seismic wave absorber under a building, an optimal ratio exists where vertical displacements and vibrations can be completely eliminated. Here we first recall the classical theoretical treatment of the positive Poynting effect, and then show experimentally how it can be reversed. Using finite-element simulations, we then investigate how the effect can be suppressed. We find that cubes always provide a reverse Poynting effect, irrespective of their material properties (in the third-order theory of weakly nonlinear elasticity).
  • Publication
    Noninvasive measurement of local stress inside soft materials with programmed shear waves
    (American Association for the Advancement of Science, 2023-03-08) Zhang, Zhaoyi; Li, Guo-Yang; Jiang, Yuxuan; Zheng, Yang; Gower, Artur L.; Destrade, Michel; Cao, Yanping; Horizon 2020
    Mechanical stresses across different length scales play a fundamental role in understanding biological systems¿ functions and engineering soft machines and devices. However, it is challenging to noninvasively probe local mechanical stresses in situ, particularly when the mechanical properties are unknown. We propose an acoustoelastic imaging¿based method to infer the local stresses in soft materials by measuring the speeds of shear waves induced by custom-programmed acoustic radiation force. Using an ultrasound transducer to excite and track the shear waves remotely, we demonstrate the application of the method by imaging uniaxial and bending stresses in an isotropic hydrogel and the passive uniaxial stress in a skeletal muscle. These measurements were all done without the knowledge of the constitutive parameters of the materials. The experiments indicate that our method will find broad applications, ranging from health monitoring of soft structures and machines to diagnosing diseases that alter stresses in soft tissues.
  • Publication
    Analysis of in vivo skin anisotropy using elastic wave measurements and Bayesian modelling
    (Springer, 2023-04-06) Nagle, Matt; Price, Susan; Trotta, Antonia; Destrade, Michel; Fop, Michael; Ní Annaidh, Aisling
    In vivo skin exhibits viscoelastic, hyper-elastic and non-linear characteristics. It is under a constant state of non-equibiaxial tension in its natural configuration and is reinforced with oriented collagen fibers, which gives rise to anisotropic behaviour. Understanding the complex mechanical behaviour of skin has relevance across many sectors including pharmaceuticals, cosmetics and surgery. However, there is a dearth of quality data characterizing the anisotropy of human skin in vivo. The data available in the literature is usually confined to limited population groups and/or limited angular resolution. Here, we used the speed of elastic waves travelling through the skin to obtain measurements from 78 volunteers ranging in age from 3 to 93 years old. Using a Bayesian framework allowed us to analyse the effect that age, gender and level of skin tension have on the skin anisotropy and stiffness. First, we propose a new measurement of anisotropy based on the eccentricity of angular data and conclude that it is a more robust measurement when compared to the classic ¿anisotropic ratio¿. Our analysis then concluded that in vivo skin anisotropy increases logarithmically with age, while the skin stiffness increases linearly along the direction of Langer Lines. We also concluded that the gender does not significantly affect the level of skin anisotropy, but it does affect the overall stiffness, with males having stiffer skin on average. Finally, we found that the level of skin tension significantly affects both the anisotropy and stiffness measurements employed here. This indicates that elastic wave measurements may have promising applications in the determination of in vivo skin tension. In contrast to earlier studies, these results represent a comprehensive assessment of the variation of skin anisotropy with age and gender using a sizeable dataset and robust modern statistical analysis. This data has implications for the planning of surgical procedures and questions the adoption of universal cosmetic surgery practices for very young or elderly patients.
  • Publication
    Voltage-controlled topological interface states for bending waves in soft dielectric phononic crystal plates
    (Elsevier, 2022-11-19) Chen, Yingjie; Wu, Bin; Destrade, Michel; Chen, Weiqiu; Horizon 2020
    The operating frequency range of passive topological phononic crystals is generally fixed and narrow, limiting their practical applications. To overcome this difficulty, here we design and investigate a one-dimensional soft dielectric phononic crystal (PC) plate system with actively tunable topological interface states via the mechanical and electric loads. We use nonlinear electroelasticity theory and linearized incremental theory to derive the governing equations. First we determine the nonlinear static response of the soft dielectric PC plate subjected to a combination of axial force and electric voltage. Then we study the motion of superimposed incremental bending waves. By adopting the Spectral Element Method, we obtain the dispersion relation for the infinite PC plate and the transmission coefficient for the finite PC plate waveguide. Numerical results show that the low-frequency topological interface state exists at the interface of the finite phononic plate waveguide with two topologically different elements. By simply adjusting the axial force or the electric voltage, an increase or decrease in the frequency of the topological interface state can be realized. Furthermore, applying the electric voltage separately on different elements of the PC plate waveguide is a flexible and smart method to tune the topological interface state in a wide range. These results provide guidance for designing soft smart wave devices with low-frequency tunable topological interface states.
  • Publication
    Non-destructive mapping of stress and strain in soft thin films through sound waves
    (Nature Research, 2022-09-17) Li, Guo-Yang; Gower, Artur L.; Destrade, Michel; Yun, Seok-Hyun
    Measuring the in-plane mechanical stress in a taut membrane is challenging, especially if its material parameters are unknown or altered by the stress. Yet being able to measure the stress is of fundamental interest to basic research and practical applications that use soft membranes, from engineering to tissues. Here, we present a robust non-destructive technique to measure directly in-situ stress and strain in soft thin films without the need to calibrate material parameters. Our method relies on measuring the speed of elastic waves propagating in the film. Using optical coherence tomography, we verify our method experimentally for a stretched rubber membrane, a piece of cling film (about 10 ¿m thick), and the leather skin of a traditional Irish frame drum. We find that our stress predictions are highly accurate and anticipate that our technique could be useful in applications ranging from soft matter devices to biomaterial engineering and medical diagnosis.
  • Publication
    Representing the stress and strain energy of elastic solids with initial stress and transverse texture anisotropy.
    (The Royal Society, 2022-10-12) Mukherjee, Soumya; Destrade, Michel; Gower, Artur L.
    Real-world solids, such as rocks, soft tissues and engineering materials, are often under some form of stress. Most real materials are also, to some degree, anisotropic due to their microstructure, a characteristic often called the `texture anisotropy¿. This anisotropy can stem from preferential grain alignment in polycrystalline materials, aligned micro-cracks or structural reinforcement, such as collagen bundles in biological tissues, steel rods in pre-stressed concrete and reinforcing fibres in composites. Here, we establish a framework for initially stressed solids with transverse texture anisotropy. We consider that the strain energy per unit mass of the reference is an explicit function of the elastic deformation gradient, the initial stress tensor and the texture anisotropy. We determine the corresponding constitutive relations and develop examples of nonlinear strain energies that depend explicitly on the initial stress and direction of texture anisotropy. As an application, we then employ these models to analyse the stress distribution of an inflated initially stressed cylinder with texture anisotropy and the tension of a welded metal plate. We also deduce the elastic moduli needed to describe linear elasticity from stress reference with transverse texture anisotropy. As an example, we show how to measure the stress with small-amplitude shear waves.
  • Publication
    Modelling brain tissue elasticity with the Ogden model and an alternative family of constitutive models.
    (The Royal Society, 2022-08-29) Anssari-Benam, Afshin; Destrade, Michel; Saccomandi, Giuseppe
    The Ogden model is often considered as a standard model in the literature for application to the deformation of brain tissue. Here, we show that, in some of those applications, the use of the Ogden model leads to the non-convexity of the strain-energy function and mis-prediction of the correct concavity of the experimental stress¿stretch curves over a range of the deformation domain. By contrast, we propose a family of models which provides a favourable fit to the considered datasets while remaining free from the highlighted shortcomings of the Ogden model. While, as we discuss, those shortcomings might be due to the artefacts of the testing protocols, the proposed family of models proves impervious to such artefacts.
  • Publication
    The Ogden model of rubber mechanics: 50 years of impact on nonlinear elasticity
    (The Royal Society, 2022-08-29) Destrade, Michel; Dorfmann, Luis; Saccomandi, Giuseppe
    We place the Ogden model of rubber elasticity, published in Proceedings of the Royal Society 50 years ago, in the wider context of the theory of nonlinear elasticity. We then follow with a short interview of Ray Ogden FRS and introduce the papers collected for this Theme Issue
  • Publication
    Nonlinear vibration and stability of a dielectric elastomer balloon based on a strain-stiffening model
    (Springer, 2022-05-23) Alibakhshi, Amin; Chen, Weiqiu; Destrade, Michel
    Limiting chain extensibility is a characteristic that plays a vital role in the stretching of highly elastic materials. The Gent model has been widely used to capture this behaviour, as it performs very well in fitting stress-stretch data in simple tension, and involves two material parameters only. Recently, Anssari-Benam and Bucchi (Int. J. Non. Linear. Mech. 128:103626, 2021) introduced a different form of generalised neo-Hookean model, focusing on the molecular structure of elastomers, and showed that their model encompasses all ranges of deformations, performing better than the Gent model in many respects, also with only two parameters. Here we investigate the nonlinear vibration and stability of a dielectric elastomer balloon modelled by that strain energy function. We derive the deformation field in spherical coordinates and the governing equations by the Euler-Lagrange method, assuming that the balloon retains its spherical symmetry as it inflates. We consider in turn that the balloon is under two types of voltages, a pure DC voltage and an AC voltage superimposed on a DC voltage. We analyse the dynamic response of the balloon and identify the influential parameters in the model. We find that the molecular structure of the material, as tracked by the number of segments in a single chain, can control the instability and the pull-in/snap-through critical voltage, as well as chaos and quasi-periodicity. The main result is that balloons made of materials exhibiting early strain-stiffening effects are more stable and less prone to generate chaotic nonlinear vibrations than when made of softer materials, such as those modelled by the neo-Hookean strain-energy density function.
  • Publication
    Plane-polarised finite-amplitude shear waves in deformed incompressible materials
    (SAGE Publications, 2022-04-07) Destrade, Michel; Saccomandi, Giuseppe
    We investigate how two finite-amplitude, transverse, plane body waves may be superposed to propagate in a deformed hyperelastic incompressible solid. We find that the equations of motion reduce to a well-determined system of partial differential equations, making the motion controllable for all solids. We find that in deformed Mooney¿Rivlin materials, they may travel along any direction and be polarised along any transverse direction, an extension of a result by Boulanger and Hayes (Quart. J. Mech. Appl. Math. 45 (1992) 575). Furthermore, their motion is governed by a linear system of partial differential equations, making the Mooney¿Rivlin special in that respect. We select another model to show that for other materials, the equations are nonlinear. We use asymptotic equations to reveal the onset of nonlinearity for the waves, paying particular attention to how close the propagation direction is to the principal axes of pre-deformation.
  • Publication
    The generalised Mooney space for modelling the response of rubber-like materials
    (Springer, 2022-03-23) Anssari-Benam, Afshin; Bucchi, Andrea; Destrade, Michel; Saccomandi, Giuseppe
    Soft materials such as rubbers, silicones, gels and biological tissues have a nonlinear response to large deformations, a phenomenon which in principle can be captured by hyperelastic models. The suitability of a candidate hyperelastic strain energy function is then determined by comparing its predicted response to the data gleaned from tests and adjusting the material parameters to get a good fit, an exercise which can be deceptive because of nonlinearity. Here we propose to generalise the approach of Rivlin and Saunders (Philos. Trans. R. Soc. A, Math. Phys. Eng. Sci. 243:251¿288, 1951) who, instead of reporting the data as stress against stretch, manipulated these measures to create the `Mooney plot¿, where the Mooney-Rivlin model is expected to produce a linear fit. We show that extending this idea to other models and modes of deformation (tension, shear, torsion, etc.) is advantageous, not only (a) for the fitting procedure, but also to (b) delineate trends in the deformation which are not obvious from the raw data (and may be interpreted in terms of micro-, meso-, and macro-structures) and (c) obtain a bounded condition number ¿ over the whole range of deformation, a robustness which is lacking in other plots and spaces.
  • Publication
    A hyperbolic framework for shear sound beams in nonlinear solids
    (Elsevier, 2021-09-14) Berjamin, Harold; Destrade, Michel; Irish Research Council
    In soft elastic solids, directional shear waves are in general governed by coupled nonlinear KZK-type equations for the two transverse velocity components, when both quadratic nonlinearity and cubic nonlinearity are taken into account. Here we consider spatially two-dimensional wave fields. We propose a change of variables to transform the equations into a quasi-linear first-order system of partial differential equations. Its numerical resolution is then tackled by using a path-conservative MUSCL-Osher finite volume scheme, which is well-suited to the computation of shock waves. We validate the method against analytical solutions (Green¿s function, plane waves). The results highlight the generation of odd harmonics and of second-order harmonics in a Gaussian shear-wave beam.
  • Publication
    Acousto-elasticity of transversely isotropic incompressible soft tissues: Characterization of skeletal striated muscle
    (IOP Publishing, 2021-07-13) Remeniéras, Jean-Pierre; Bulot, Mahé; Gennisson, Jean-Luc; Patat, Frédéric; Destrade, Michel; Bacle, Guillaume
    Using shear wave elastography, we measure the changes in the wave speed with the stress produced by a striated muscle during isometric voluntary contraction. To isolate the behaviour of an individual muscle from complementary or antagonistic actions of adjacent muscles, we select the flexor digiti minimi muscle, whose sole function is to extend the little finger. To link the wave speed to the stiffness, we develop an acousto-elastic theory for shear waves in homogeneous, transversely isotropic, incompressible solids subject to an uniaxial stress. We then provide measurements of the apparent shear elastic modulus along, and transversely to, the fibre axis for six healthy human volunteers of different age and sex. The results display a great variety across the six subjects. We find that the slope of the apparent shear elastic modulus along the fibre direction changes inversely to the maximum voluntary contraction (MVC) produced by the volunteer. We propose an interpretation of our results by introducing the S (slow) or F (fast) nature of the fibres, which harden the muscle differently and accordingly, produce different MVCs. A natural follow-up on this study is to apply the method to patients with musculoskeletal disorders or neurodegenerative diseases.
  • Publication
    Bending control and stability of functionally graded dielectric elastomers
    (Elsevier, 2021-01-18) Su, Yipin; Ogden, Ray W.; Destrade, Michel; Irish Research Council
    A rectangular plate of dielectric elastomer exhibiting gradients of material properties through its thickness will deform inhomogeneously when a potential difference is applied to compliant electrodes on its major surfaces, because each plane parallel to the major surfaces will expand or contract to a different extent. Here we study the voltage-induced bending response of a functionally graded dielectric plate on the basis of the nonlinear theory of electroelasticity, when both the elastic shear modulus and the electric permittivity change with the thickness coordinate. The theory is illustrated for a neo-Hookean electroelastic energy function with the shear modulus and permittivity varying linearly across the thickness. In general the bending angle increases with the potential difference, and this enables the material inhomogeneity to be tuned to control the bending shape. We derive the Hessian criterion that ensures stability of the bent configurations in respect of a general form of electroelastic constitutive law specialized for the considered geometry. This requires that the Hessian remains positive. For the considered model we show that the bent configuration is stable until the voltage reaches the value for which the cross-section of the bent configuration forms a complete circle.
  • Publication
    On the thermodynamic consistency of Quasi-linear viscoelastic models for soft solids
    (Elsevier, 2020-12-15) Berjamin, Harold; Destrade, Michel; Parnell, William J.; Irish Research Council
    Originating in the field of biomechanics, Fung's model of quasi-linear viscoelasticity (QLV) is one of the most popular constitutive theories employed to compute the time-dependent relationship between stress and deformation in soft solids. It is one of the simplest models of nonlinear viscoelasticity, based on a time-domain integral formulation. In the present study, we consider the QLV model incorporating a single scalar relaxation function. We provide natural internal variables of state, as well as a consistent expression of the free energy to illustrate the thermodynamic consistency of this version of the QLV model. The thermodynamic formulation highlights striking similarities between QLV and the internal-variable models introduced by Holzapfel and Simo. Finally, the dissipative features of compressible QLV materials are illustrated in simple tension.